Structure and Dynamics of Saturn’s Rings

Larry W EspositoLASP, University of Colorado

29 March 2018

Saturn Equinox 2009

• Oblique lighting exposed vertical ring structure and embedded objects

• Rings were the coldest ever• Images inspired new occultation and spectral

analysis• Steady progress and new discoveries continue:

More complex, time variable

Mimas Casts a Long Shadow at Equinox

Aggregates form at outer B ring edge

Sub-km structure seen in wavelet analysis varies with longitude

• Wavelet analysis from multiple UVIS occultations is co-added to give a significance estimate

• For the B ring edge, the significance of features with sizes 200-2000m shows maxima at 90 and 270 degrees ahead of Mimas

• For density waves, significance correlated to resonance torque from the perturbing moon

‘Straw’ seen between density wave crests

Solitary wave propagating through A ring?

Daphnis Edge Wake

Clumps Form

Ring Edge Shears and Separates

Predator-Prey model of Moon-triggered Accretion?

Predator-Prey Equations for Ring Clumping

M= ∫ n(m) m2 dm / <M>; Vrel

2= ∫ n(m) Vrel2 dm / N

dM/dt= M/Tacc – Vrel2/vth

2 M/Tcoll

[accretion] [fragmentation/erosion]

dVrel2/dt= -(1-ε2)Vrel

2/Tcoll + (M/M0)2 Vesc2/Tstir

[dissipation] [gravitational stirring]

- A0 cos(ωt) [forcing by streamline crowding]

Not included: Separate evolution for small and large particles; Tidal forces.

Why this simplified model?

• N-body simulations can’t include all the relevant physics and consider long azimuthal and time scales

• Predator-Prey model gives some useful intuition about equilibrium points, stability, and asymptotic behavior

• Provide direction for more detailed models and key observations

Strength regime, the particles have reached equilibrium, Sigma forcing

What have we learned?

• Moon-forced streamline crowding can cause temporary aggregates, out of phase with the moon

• Disk instability is needed to produce clumps on the orbital time scale

• Perturbations by Pan and Daphnis may grow large enough through disk instability to tear apart lanes in the nearby rings

Post-Equinox View• Cassini Equinox observations show Saturn’s

rings as a complex geophysical system, incompletely modeled as a single-phase fluid: clumps evident; particles segregate by size; viscosity depends on shear; shear reverses in perturbed regions; rings are far from equilibrium in perturbed regions

• Self-gravity causes wakes, viscosity, overstabilty and local aggregate growth

• Larger fragments: seeds for growth

Ring dynamics and history implications

• Moon-triggered clumping at perturbed regions in Saturn’s rings creates both high velocity dispersion and large aggregates at these distances, explaining both small and large particles observed there.

• A simple ecological model can give us some insight• This confirms the triple architecture of ring particles:

a broad size distribution of particles; aggregates into temporary rubble piles; coated by a regolith of dust..

• Cassini results show the rings are much younger than the Solar System, but they may replicate processes in planet formation

Backup Slides

Are Saturn’s Rings Young or Old?• Voyager found active processes and short lifetimes:

we concluded the rings were created recently. • Because it is highly unlikely a comet or moon as big

as Mimas was shattered recently to produce Saturn’s rings, we ask: Are we very fortunate?

• Cassini observations show a range of ages, some even shorter!

• Esposito etal (1983) considered the Voyager mass estimate a lower limit

• Conversely, density waves in the B ring show less massive rings (Hedman and Nicholson 2016)

• Cassini confirms earlier dust estimates. Because meteoritic dust pollutes rings: Less massive rings must be very young: tens of millions of years

Why are Saturn’s rings so active and dynamic?

1. Granular material: Particle-to-particle collisions dominate;Kinetic, not fluid description needed; Stresses are strikingly inhomogeneous;Fluctuations large compared to equilibrium.

2. Strongly forced by resonances:Non-linear response to moon forcing; Thresholds lead to persistent states.

Pred-Prey Model:

Saturn’s B ring is the brightest and most opaque

Conclusions• Ring structure shows clumping in perturbed regions: Moon forcing

triggers aggregation. Lewis & Stewart simulations show streamline crowding and reduction of the velocity dispersion downstream from the satellite

• The structure forms rapidly, on orbital time scales, out of phase with the moon, and is reduced at the next moon passage (shown by wavelet analysis, straw, gaps, statistics)

• A simple two parameter model for the aggregate mass and velocity dispersion captures the key parts of the dynamics. This ‘Predator-Prey’ model allows ecological analogies, and can be directly related to the pendulum and the Duffing ‘Moon-beam’ oscillator, both well-studied non-linear systems

• This model shows the fixed points, their stability and the phase plane response from driving the system. Cell-to-cell mapping techniques transform the trajectories to a Markov process

Conclusions (2)• Stochastic collisions lead to larger, compacted,longer-lived aggregates• The Predator-Prey model can be adjusted to Goldreich’s ‘Two-Group’

model by specifically defining the small and large particles as the ring particles and the clumps

• In this approximation, we can calculate the fixed points and the growth rates in the driven system: We find that growth by aggregation is too slow to explain the excess structure observed in between density wave crests

• Gravitational disk instability can act on orbital time scales; We use Toomre’s stabilty parameter Q to estimate the growth rate for clumps

• We can achieve rapid growth by reducing the increasing the surface mass density, decreasing the velocity dispersion or decreasing the shear. This last is like ‘swing amplification’

Conclusions (3)• We can identify the fixed points with self-gravity wakes and those of

the longer-lived clumps with the Equinox objects, straw and the larger kittens

• Disk instability is a violent one: as the wavelength of the disturbance shrinks to zero, the growth rate grows without limit – a cold, zero-thickness disk disintegrates on small scales in an arbitrarily short time (Binney and Tremaine, 2008, p495). This can explain the gaps ripped open at the edges of the Encke and Keeler gaps (Albers 2012)

• The ring clumpiness means that we underestimate the ring mass and viscosity, both dominated by the aggregations, which have a different physical nature, like a phase change in a fluid (Tremaine 2003): this could give a different dispersion relation for density waves, leading to an underestimate of the ring mass from density wave analysis.

• A larger ring mass means that Saturn’s rings could be ancient

Rare accretion can renew rings

• Solid aggregates are persistent , like the absorbing states in a Markov chain

• Even low transition probabilities can populate these states

• These aggregates– shield their interiors from meteoritic dust

pollution – release pristine material when disrupted by an

external impact